You shim up the cabinetry at install to accommodate the flooring you select. Flooring need not be under all of the perimeter. This is routine in new builds, where flooring is generally near the end of a build. Flooring is actually one of the first selections, in conjunction with appliances, Cabinetry is derived based on appliance need and locale.

Pecking at a kitchen re do is not wise. it takes planning at the start, and no amount of planning is too much planning. Saving resources, until the entire plan can be accomplished is usually your best result.

You have nowhere near enough information to provide good answers. Upload pictures of the current condition, along with a neat and scaled drawing that includes every possible inch of everything........clearly noted. Add it all in a comment , this same thread.